Let’s begin by distinguishing the role of head of state from the role of head of government.
The minimum function of a head of state is to symbolise the unity of the nation and to exercise that symbolic role in representative, ritual and ceremonial ways that help keep the body politic together. It is difficult for a partisan figure to symbolise national unity; by definition he or she has supported one political viewpoint and opposed others in the past and his or her one-time opponents may be reluctant to give such a figure the benefit of any doubt.
The function of a head of government by contrast is to serve as head of the executive, that is to say to be in charge of getting the work of government done. In a democracy this role does not require unanimous support; customarily it is supported by a majority or the largest minority, though there are exceptions to this principle.
Some constitutions combine the two roles, though not always comfortably. These are known as executive presidencies. They range from what are effectively autocracies to highly polarised and divided democracies, though sometimes you do find an individual capable of transcending party animosity and becoming a genuinely national leader. Where such a figure fails to emerge there is no obvious alternative as a national unifying force. The role may fall to an independent judiciary or sometimes society may suffer a loss of cohesion.
For this reason there is merit in keeping the roles of head of state and head of government separate. Since the head of government is elected it is probably best if the head of state is not, because you don’t want a struggle between rival mandates.
The virtues of a monarchical system include new incumbents having experienced an apprenticeship, sometimes accepting an increasing share of royal duties as the reigning parent ages; it is rare for a complete ingénu to ascend the throne not knowing how to behave. As long as the restrictions upon this role are understood by all it is nowadays rarely controversial.
By contrast it is quite common for incoming political heads of government to lack experience or knowledge, resulting in errors of policy in early years.
You probably couldn’t introduce a monarchical system today, but that doesn’t mean it’s no longer fit for purpose or could easily be improved upon. Remember the guiding principle of management by exception - ‘If it ain’t broke, don’t fix it!’